Find the measure of the angle given below
2 answers:
<h3>Answer: 25 degrees</h3>
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Explanation:
You subtract the arc measures, and then divide the result in half.
The order of subtraction is larger arc minus smaller arc.
Angle ABC = ( (arc AC) - (arc DE) )/2
Angle ABC = (90 - 40)/2
Angle ABC = 50/2
Angle ABC = 25 degrees
Side note: segments AB and BC are considered secant segments.
The next section will go into a more in-depth look. The next section is optional.
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Alternative Method:
Refer to the diagram below. I've added points D and E to the diagram.
Based on those points, let
- x = measure of minor arc CE
- y = measure of minor arc DA
Note how inscribed angle CAB subtends the arc CED. The measure of arc CED is 40+x. Half this is (40+x)/2, which is the measure of inscribed angle CAB. Through similar logic, inscribed angle ACB is (40+y)/2.
Also, note how the four arc pieces of the full circle are: AC = 90, EC = x, DE = 40, and AD = y.
The four arc pieces must add to 360 as they form a full circle
AC+EC+DE+AD = 360
90+x+40+y = 360
x+y+130 = 360
x+y = 360-130
x+y = 230
y = 230-x
From this, we can rewrite the expression for inscribed angle ACB
angle ACB = (40+y)/2
angle ACB = (40+230-x)/2
angle ACB = (270-x)/2
Now because the interior angles of a triangle must add to 180, we can then use this fact to find angle ABC
(angle ABC) + (angle CAB) + (angle ACB) = 180
angle ABC = 180 - (angle CAB) - (angle ACB)
angle ABC = 180 - (40+x)/2 - (270-x)/2
angle ABC = 180 - 40/2 - x/2 - 270/2 + x/2
angle ABC = 180 - 20 - x/2 - 135 + x/2
angle ABC = 25 degrees
Ultimately the x terms cancel out when we combine the -x/2 and +x/2 together.
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